A Modelling Method for the Behaviour of Converters Operating in Control Loops

A MODELLING METHOD FOR THE BEHAVIOUR OF CONVERTERS OPERATING IN CONTROL LOOPS R. Prajoux, J. Jalade, J .C. Marpinard and J. Mazankine Laboratoire d'Automatique et d'Analyse des Systemes du Centre National de la Recherche Scientlfiquc, 7, avenue du Colonel Roche, 31400 Toulouse, France

ABSTRACT The models available to study the dynamic behaviour of control loops including power convertors can be separated in two classes those very simplified or those very sophisticated. In this paper, we give a method which takes place between the two classes and we obtain non linear continuous models for large signal or linear continuous models for small signal. Some examples show the good agreement between the corresponding results and those obtained by using more accurate methods. 1.

INTRODUCTION

In this paper, we are primarily interested in the modeling of some convertors (especially AC-DC and DC-DC convertors) with a control point of view. A lot of work has been done in this field. This work was summarized in the survey-paper by LAGASSE-PRAJOUX at the 1st I.F.A.C. Symposium /-1 It appears that the various analytical approaches lead to the following clas;ification : they are very easy to use and permit the synthesis - linear continuous models of correcting devices,_but they aEe very approximate regarding the actual behaviour of the system, / 2,3,4,5,6 / ; - discrete line~r mod;ls they ~escribe accurately the small signal behaviour taking into account the discrete nature of the system, but the synthesis is not straight forward /-7,8,9,10 I ; - discrete non-linear model; this class is not very extensive, due to the great complexity involvedL-ll,12,13_1.

7.

In this paper, we give a method which is in fact a compromise between complexity and accuracy. Basically, one obtains a non-linear continuous model. This means that the non-linear properties of the actual system are preserved and that an approximation is made when passing from discrete to continuous. Furthermore, it is clear than a linear small-signal model can be derived from the previous one. The method consistsfirst in dividing the electrical circuit representi~ the power part of the convertor in two or more blocks. Each block has an invariant electrical structure but the interconnexions between the blocks are variable according to the state of the on-off components. Second, we determine which block (s) delivers the signal (s) applied to the control-loop (we call it the "fixed part"). Third, provided that some conditions are fulfilled, we replace the other blocks and their variable interconnexions by some equivalent blocks connected to the fixed part. We obtain thus a non-linear invariant circuit, i.e. a non-linear continuous model. The equivalence is based on averaging techniques and the method provides then a systematic way of achieving average-techniques and preserving the non-linear properties of the actual system. 2.

GENERAL STUDY

Let us consider an electrical circuit consisting in sources, passive components and switches (fig. 2.1). The switches represent the active on-off components of the a~tual convertor.

53

R. Prajoux et aZ

54

-------,~cr--,.-------I

I I

I

~

I

sub_circuits

L --[I~

I

I

-IJ

O-----'L.

I I

J

------- -. I

I I

I

~

cr"""""<>-i

_________ 4

Fig. 2.1 Elec"trical circuit with switches Switches provide variable interconnexions between sub-circuits. According to the state of those switches, the circuit has not an invariant topology but j possible configurations. Let Z be a state vector of the circuit such that

(2-]) ~c.

~

i

L

.

V-c 2

Z

•••••••••••

iL ... 2

J

T

being the voltages accross the capacitors and the currents through the inductors.

~

We suppose that Z(t) is continuous at switching instants and that there is not capacitive loopS nor inductive nodes. The circuit may then be represented by differential state equations : Z = A Z + B U k k

(2- 2)

{

wi th

]

~

k

~

j

Equation (2-2) may be solved step by step and taking for each initial value the final value of the preceding evolution. An alternative approach is to study the manner in which components of the state vector Z interact with each other for all the structures in the range [I, . . . . j] Connections between components of A k

2.] -

We say that the component z. of Z is not connected to the component z. ~ J th k structure if (a. ')k = 0 ~J

, (a. ')k being an element of Ak · ~J

One can prove that, (a. ')k J ~

=

for the

in this case

0

Then, in matrix A , k di agonal.

the zeros appear to be symmetrical with respect to the first

2.2 - Variable and fixed parts Now, we are going to separate the system in two sub-systems, represented by state vectors X and Y. Then:

Z = [X

Y]

T

The system X is called the fixed part and is such that -

its structure is invariant the sources U are inside the outputs S depend on X

Y is called the variable part and is such that -

its structure is invariant the sources U are not inside

behaviou~

Modelling method for the

55

of converters

the outputs S do not depend on Y.

Then

(2- 3)

[:l

[ [M

D


tn(k-I)

2.3 -

+

~

MX

S

[:J 1t1 at1 C k

A

[I,

Ykf:

t nk

....

jJ,

Vn

Equivalent system

.

Let us consider the system X above. We get ( 2- 4)

X = AX + B u

+

~

y

Y behaves like an input applied to system X.The response related to this input is: ( 2 - 5) X ( t)

i

=

r 0(t

-

~

rt) .

to

.

with

Y (A).d (\.

o(t)

e

=

At

Since some components of X may be not connected to some components of Y, we may wri te : (2- 6)

and

:

=

(2-7 )

(['fJ(t-1)~'f2 (t _A)J.r~k_(~~.dA L@J

)t;o

=

'I.

(l:~(t_(\). CLK({\.).drt.

)to

The above decomposition in expression (2-6) may be valid for any k, provided that the decomposition is carried out in the worst case. Now, let us assume that:

(2-8) Then : (2- 9)

f,(t_~) ~

constant,""'V-A.

1t.p,(t_~).aj;.(~).d{\' ~

E

[t

01

tJ

t

'f1

rCOkP.).dlt

t~

to

Let us consider a complete cycle between the switching instants i.e. a time interval involving the sequence of the j We get then: tn .. .,

i J: o

(t-~).

(2-10)

t.... +

Let be

1

t"..../\

(2-12) t

(2-10)

"2 a 2 ({\).

be considered as

(2-11)

n+]

-t

n

C\')

can be written as

-<') .dd

. u(~)d{\.

t

Cn"

The sequence of vector a(;\)

Then,

B

t and t + , n n l possible structures.

d (l

+ ••.•

rn..

1

+

'1

a . ( . J

t" ( J-1)

~) • d),

the evolution of a single

]

R. Prajoux et aZ

56

~(t

(2-13)

n+

1- t

n

). X + n

which is the solution of •

(2-14)

X

=

i

tn~1

+f1

~ ( t - ~ • B • U «(\..) • d C\..

l:"..,

.~.(t n+ ) - tn)

IIJ

AX + BU + a

'-P

Then, provided that within a time interval D..t, the 1 (t) is almost constac.t behavior of the fixed part can be approximated by replacing the variable part by an equivalent input

a.

Let us consider now the variable part

.

(2-15)

Y = DY + Fk·U +

Ilk'X

We ge t

t:

( 2 - 1 6) with

:

\fJ ( t -

Y ( t)

'fI

=

(t)

e

Assuming that ( 2 - 1 7)

X ( t)

Dt

t ). Y + ( l}' ( t - (1..) • F • U «\) . d C\o 0 )b k. o

1. + Dt + ••••••

: ~

Xn ,

for t n

<

t

<

t n +1

we get I.JJ(t 1- t ).Y + T n+ n n

We can note

the above expression by :

( 2-18)

...•.

t n +), Xn )

Let us define a continuous variable y such that (2-19)

y(t ) n

=

Y n'

-V-n

We have

y

Y

-Y

t

-t

n+1

~

n+ 1

n t

n

n+J

-t

n

By denoting (2-20)

et'n

= [tn ,

(2-21)

y

D.y + G('(;,X)

, •••.•. t + ] T n 1 n1 the above expression may be written : =

n

Combining (2-6), ( 2-22)

t

(2-12)

1

, including t

n+ 1 t n

and integrating (2-16)

into the function G. gives

an expression in the form

'V

a

n

or (2-23)

N

a

F(

~

,X,y), using continuous variables.

Among the instants grouped into vector~ , some are given by the control system, say 'CE: , others are fixeQ, say_ ~F and the remaining depend on the state vector, Le. on X and Y L 17 _I. Then

:

(2-24) Combining (2-14), (2-21), (2-23)

and (2-24)

gives

the set of equations

57

Modelling method for the behaviour of converters

(2-25)

X Y

AX + BU +

~

F( ~,X,y)

<{;

ca

Dy + G(~,X)

=

C , ~F" F

+ C , T(X,y) T

+ C , qfe: E

The system (2-25) is non-linear since the functions G,F and T are non-linear. The inputs are U and ~E ' but U is known a priori and has a predetermined value, then ~E: is the true control input, To conclude this paragraph, we can write that ~

provided that some conditions are fulfilled, the non-linear but piecewise linear system (2-3) can be represented approximately by the non-linear continuous system (2-25) - the action of the variable part on the fixed part is approximated by fictitious voltage or current sources equal to the average of the actual signals ; these fictitious sources are controlled by an equivalent continuous system. 3, APPLICATION TO A CLASS

or

DC-DC CONVERTORS

In order to illustrate the above theoretical concepts, we are going now to deal with an application to a class of DC-DC convertors used especially in aerospace instrumentation (the so-called buck, boost and buck-boost convertors), although the method of paragraph 2 is relevant to a much wider class, The circuit diagram of a boost convertor is given as an example on fig.

L

.

IL

3. I.

D ~

r

+ E

T

R

C

Vs

Iv

o Fig, 3,1

BOOST CONVERTOR

The two other convertors differ only from the first one by the topology of the 3 devices L,D,T /-14,]5,]6 7 and, thus, the block-diagram of fig. 3,2 can be given for the whole class:

R. Prajoux et aL

58

1

-

-

-

I

------t

-

-

-

....

"' . . 1.

---

Inductor L Transistor T Diode D For 3 types of unit BUCK, BOOST,

r-

I

II

--=pu7::t=:-- -

'I

I

BUCK-BOOST

1

V

,

power

I I

stage

RI

c

i

~(p)

t:;F'

_

1 J-

_J

-----J

I

Vs

~G~M~-~

I

I

I I r

Vs

I- __ ~- --

•

,-----

t

I

-I

-

1

R

II -------...-----PW-Modulator

1_ _

Amplifying and correcting device

+ V

Control

Figure 3.2 - Block-Diagram for the class of DC-DC convertors

3.1 Modeling of the class The given class is a special case for systems (2-3) and significant simplifications can be carried out. - The fixed part consists in the output network and the source E. - The variable part consists in the inductor L only. part : .X = AX for+ CkYthe , fixed since E is not

Then,

connected.

We have X=v, the voltage accross the capacitor C, of an actual capacitor), and then: (3-1)

~

= [ - (r:R) C ] v

+ C i [r:R

i]

(r represents the series resistance

iL

The scalar C. can take the values 0 or 1 and the quantity Cii is actually the L current denoted i on fig. 3.2 : (3-2)

i

= Ci.i

(3-3)

0(t)

L It is easy to verify that the validity conditions of paragraph 2 are fulfilled. = e

-

1

(r+R)C

t

, then e

___ 1- T (r+R) C

v n+1 -'" v n

since C must have a value great enough to provide a good filtering effect. This leads for the fixed part to the equivalent circuit diagram of fig. 3.3. The current generator i(t), whose Laplace transform is I(p), is a pulse current generator. This generator can be replaced by an equivalent generator Yet) ] r(p). The current i(t) depends on the type of convertor, the conditions of operation and on some variables.

Modelling method for the behaviour of converters

r-------I-;::=--......,- 'V

I

+- J:

i (p)

5( P

Vs( p)

- - --L,-Output

'-------If----' -

Figure 3.3 -

S9

-

network

Equivalent circuit of power stage

By definition IV

(3-4)

i ( t)

The figure 3.4 shows two typical waveforms for the current i , fig. 3.4a being the case of the continuous conduction and f. 3.4b the discontinuous conduction.

'L 'M I

m

IF

1

Fn

I

I

I

ONi

TOFF

I F('n+1)

I I

I

m(n+1)

I

I I I

T

I

I

I

I

I

I

I

I

i

:

I I I

I Mn+--""*-----T---

I I I

I

I

- T - -..... ~

.. Fig.

3.4

Typical waveforms for i

L

N

The current i must be computed - the supply voltage E - the output voltage Vs ::!. v -

from the value of i

L

and then from

the inductor L the time parameters T and TON T

=

= OFF

energy storage time T-T

ON

Instead of computing '" i directly from the waveform of I , we carry out an indiL rect computation hy using the variables I , I and I appearing on fig. 3.4. Mn mn Fn The reason for doing this will appear later on. Thus we define the equivalent continuous variables i , i and iF which has the M m same values than I , I and IF respectively, at the signlficant discrete insM m tan ts. Then, we must compute 'V

(3-5)

i

=

f(im,iM,iF,tON,T)

i

Direct computation of would have hidden the true operating conditions of the convertor (for example the transition between continuous and discontinuous

R. Prajoux et

60

conduction due to the transition of i

a~

through the value zero).

m

Knowing i and tON' it is possible to compute i and iF' Consequently (3-5) M in fact i~ the form

The interpretation of equation (2-21) I -I ij,i (T) m(n+ 1) mn m fV (3-7) i T

m

is

is, in continuous conduction I

Fn

-I

mn

T

T

It is important here to emphasize that the approximation involved in expression (3-7) does not i1l1ply that ij,im(T) should be small compared to i , but only that m tON varies slowly from one period to the following one, i.e.

«

( 3-8)

In discontinuous conduction,

{ .. i

(3-9)

1

mn

m

, V

o o

1

the set (2-25)

is degenerated and

y~O,

i.e.

n

The explicit appearan~e of. i and i has a further advantage. It is possible to m M introduce extra non-l1near1t1es, such as the current limitation in the switch. 3.2 -

Example:

the buck convertor

In order to derive for this example a very general model, we shall consider that ~ON and tOFF are two completely distinct imputs. Consequently we shall obtain a model valid for, either: - fixed-frequency convertors (T=constant) - or variable-frequency convertors (T!constant). The latter case is very interesting, frequency is variable.

as most theoretical methods fail if the

Contin"uous conduction. Let us consider a "buck" convertor (fig.

3.5).

B5(P) Figure 3.5 -

BUCK CONVERTOR

The electrical structure of the buck is such that i(t) = iL(t),

"tt

In the case of a current limitation, 3.6a.

the waveform for iL(t)

It is easy to establish that 'V

(3-10)

i ML ( t) + i F ( t)

i(t)

2

(i

(t)-i (t))2 M ML

iM(t)-im(t)

+

is shown on fig.

61

Modelling method for the behaviour of converters N

Discontinuous conduction, the relation: 'V

i

DISC

=

'V

i

CONT

+

Instead of computing directly the value of i, we use

jt

J!being the area appearing on fig.

3.6b.

'V

=

iCONT(t)

t

fictitious value /,--

PE'N od n ...

~A

____..,

iL

I

I

limitation

limitation

IF{n)

I

••

pE'riod n +1

a) continuous conduction

b) discontinuous conduction

Figure 3.6 - Inductor current The above relationship is very interesting in building a simulation block diagram since it is then possible to take into account the changes from a conduction mode to the other without any initialisation problem. Block-diagram of the buck convertor. The block-diagram of the buck convertor obtained with the above computations is shown on fig. 3.7 . This block-diagram is valid for any amplitude of the signals, provided that the inputs tON and tOFF vary slowly. The conduction mode is controlled by a flip-flop whose state changes according to the sign of the variables i m or iF' tofF

Figure 3.7 - Large-signal block diagram of the buck convertor

mode of conduction

R. Prajoux et aZ

62

(X}40~-1 I",o + I

M0

Figure 3.8 - Small-signal model for the buck in continuous conduction -Small-signal model Near a given operating point, it is possible to differentiate the out put value of each non-linear block and to get, then, a linear model valid, for small-signal only. This model, for the buck, is shown on fig. 3.8. After reduction, the linear block-diagram leads to the model of fig. 3.9, in the case of a fixed-frequency operation.

1

Lp

Figure 3.9 - Small-signal model for fixed-frequency operation

63

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1"1 •• "''''OH .. ''t

v. ,

l.for'JIJIH.-UI ,'. \.'.' ,11)t,-II1

·gt -'11 ~: 'o' J v!. - 'J I

t.~, ...

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'~V \;~

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v, •"'''

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Figure 3.10 - Example of a CSMP simula tion Improv ement of the model s range for which the above model s In order to extend signi fican tly the freque ncy T/2 must be insert ed in the delay a that L-17 are valid, it has been shown This input conne xions. L 2 sed has propo STEN BJ~RE than value delay T/2 has the same

7

7.

Examp les using above model s 3.7 is suffic iently compl ex to The large- signa l model s such as the one on fig. theles s, th~ are very useNever . study tical analy ult lead to extrem ely diffic mming langua ges like the progra using ful to carry out simul ations , espec ially compu tation time, accorthe in ement improv great a then, is, IBM-CSMP. There one conve rtor period within steps ration integ ding to the fact that only a few are neede d. 3.10 for a buck conve rtor An examp le of a CSMP simul ation j s shown on fig. initia lisati on. A hybrid the on which can be stable or unsta ble, depen ding 3.11 and a very good agreefig. on shown is rtor conve l actua simul ation of the system occur at a frethe of ns l~tio ment can be observ ed althou gh the oscil 1/4T. as high quency as

Inducto r curren t Figure 3.11 - Hybrid Simula tion - Modula tor Sawtoo th -

R. Prajoux et al

64

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M o d U Iu50 (VI:) 50

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Figure 3-12 - Bode diagram for a buck convertor

Dn fig. 3.12, a Bode diagram obtained from the small-signal model of the buck is shown. The corresponding convertor has the step response on fig. 3.13a. Using the linear small-signal model, the system can be corrected and we get the response on fig. 3.13b. This response is as good as possible and illustrate the satisfactory accuracy of the model.

0.0': ECH N

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o

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Fig 3.13 - 50% step, buck response

CONCLUSION We have described a method to obtain systematically an approximate model for electrical convertors. The model is continuous and non-linear. It is especially suitable to carry out digital simulations with low computation time but very good accuracy. The linear model which can be derived from the previous one is very useful to synthesise a correcting device to obtain a satisfactory transient b~haviour. The method is

then,

a good compromise between complexity and accuracy.

65

Modelling method for the behaviour of converters REFERENCES

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L2 7 L

J.LAGASSE, R. PRAJOUX Behaviour of control systems including controlled convertors, especially rectifiers : a review of existing theories Symposium IFAC "Control in power electronics and electrical drives"· (1974). N.A.BJaRESTEN The static convertor as a high-speed power amplifier Direct Current, 8,154 (1963).

37-

L 47L5

/ -

L6 7-

convertors Conference Atlantic

G.W.WESTER Linearized stability analysis and design of a fly-back DC/DC boost regulator IEEE Power Electronics Specialists Conference, Caltech. (June 1973~ R.D.MIDDLEBROOK A continuous model for the tapped-inductor boost-convertor IEEE Power Processin Electronics Specialists Conference, Los Angeles (1975 A.K.LAHA, K.E.BOLLINGER Power stabiliser design using pole-placement techniques on approximate power system model Proc. lEE 122 (Sept. 75),

L

7

7-

L8 7 -

L9 7L 10 7 L

11

7-

L 12 7 L

13

7-

G.GIRALT, J.LAGASSE,Y.SEVELY Sur la stabilite locale de la boucle de reaction de dispositifs lateurs de tension a redresseurs controles C.R. Academie des Sciences PARIS ~2 mars 1962l

regu-

R.PRAJOUX Etablissement d'un modele pour le comportement local d'un redresseur polyphase utilise en tant qu'amplificateur de puissance a reponse rap ide C.R. Academie des Sciences PARIS (22 Sept. 1969~ H.BUHLER Investigation of a rectifier regulating circuit as s y s te m 4th IFAC Congress, Wars zawa, (June 1969).

a sampled data

D.SCHRODER Analysis and synthesis of automatic control systems with controlled conve rtors 5th IFAC Congress, Paris, (June 1972). R. VALETTE, R.PRAJOUX, A.GIRAUD Systemes a thyristors :comparaison entre un mOdele neur et un modele recurrent AFCET, RAIRO Review, J3, (1974).

avec echantillon-

H.FOCH Applied study of a fast commutation circuit with an auxiliary supply IFAC Symposium "Control in Power Electronics and Electrical Drives" Dusseldorf (1974), F.LEE, T.G.WILSON, S.FENG Analysis of limit cycles in a two transistor saturable core parallel inverter IEEE Transactions an Aerospace and Electronics Systems, AES 9 (July 1973).